Question

If the function $f(x)=2 x^{3}-9 a x^{2}+12 a^{2} x+1[a>0]$ attains its maximum and minimum at $p$ and $q$ respectively such that $p^{2}=q$, then $a$ is equal to
(A) 2
(B) $\frac{1}{2}$
(C) $\frac{1}{4}$
(D) 3

Answer 1

Ans : (A)
Hint : $f^{\prime}(x)=6 x^{2}-18 a x+12 a^{2} \Rightarrow f^{\prime \prime}(x)=12 x-18 a \Rightarrow f^{\prime}(x)=0 \Rightarrow x=a, 2 a$
$f^{\prime \prime}(a)<0 ; p=a$ (maximum)
$\mathrm{f}^{\prime \prime}(2 \mathrm{a})>0 ; \mathrm{q}=2 \mathrm{a}$ (minimum)
$a^{2}=2 a ; a(a-2)=0, \quad a=2$

Answer 2

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Answer 3

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